Advertisements
Advertisements
प्रश्न
If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
विकल्प
\[\frac{ab}{2 (b - a)}\]
\[\frac{ab}{b - a}\]
\[\frac{3 ab}{2 (b - a)}\]
none of these
Advertisements
उत्तर
\[\frac{3 ab}{2 (b - a)}\]
Let the A.P. be a, a+d, a+2d........a+nd.
Here, let d be the common difference and n be the total number of terms.
\[a_1 = a, \]
\[ a_2 = b\]
\[ \Rightarrow a + d = b\]
\[ \Rightarrow d = b - a . . . . . \left( 1 \right)\]
\[ a_n = 2a\]
\[ \Rightarrow a + \left( n - 1 \right)d = 2a\]
\[ \Rightarrow \left( n - 1 \right)d = a\]
\[ \Rightarrow d = \frac{a}{n - 1} . . . . . \left( 2 \right)\]
Given:
From equations \[\left( 1 \right) \text { and } \left( 2 \right),\] we have:
\[\Rightarrow \frac{a}{n - 1} = b - a\]
\[ \Rightarrow \frac{a}{b - a} + 1 = n\]
\[ \Rightarrow \frac{a + b - a}{b - a} = n\]
\[ \Rightarrow \frac{b}{b - a} = n\]
Now, sum of n terms of an A.P.:
\[S = \frac{n}{2}\left\{ a + a_n \right\}\]
\[ = \frac{n}{2}\left( 3a \right)\]
\[ = \frac{3ab}{2\left( b - a \right)}\]
APPEARS IN
संबंधित प्रश्न
Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.
The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.
Find:
10th term of the A.P. 1, 4, 7, 10, ...
If the sequence < an > is an A.P., show that am +n +am − n = 2am.
Find the 12th term from the following arithmetic progression:
3, 5, 7, 9, ... 201
An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.
Find the sum of the following arithmetic progression :
50, 46, 42, ... to 10 terms
Find the sum of the following arithmetic progression :
3, 9/2, 6, 15/2, ... to 25 terms
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of all integers between 100 and 550, which are divisible by 9.
The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
If a, b, c is in A.P., prove that:
a2 + c2 + 4ac = 2 (ab + bc + ca)
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.
A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
A man starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be
In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
Mark the correct alternative in the following question:
Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
Show that (x2 + xy + y2), (z2 + xz + x2) and (y2 + yz + z2) are consecutive terms of an A.P., if x, y and z are in A.P.
The sum of terms equidistant from the beginning and end in an A.P. is equal to ______.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
The sum of n terms of an AP is 3n2 + 5n. The number of term which equals 164 is ______.
If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is ______.
